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Full text of "Scientific Papers - Vi"

HYDKODYNAMICAL NOTES
17
method of obtaining it. The results are in harmony; but the fact is not immediately apparent, inasmuch as Stokes expresses the motion by means of the velocity-potential, whereas here we have employed the stream-function.
That the subtraction of £«r2 makes (10) an harmonic function shows that the series multiplying r2 can be summed.    In fact
sin(n7rO/ot)    _ cos (2(9 — a)     1 vnr (nV2 — 4a2) ~    2 cos a         2 '
o 2
so that
cos (20 -a)         ^rn^asinn7T0/a
=-^ - -+8a22 — — -^ — -r1-2 cos a                    n7r(n~7r2—4<a2)
(11)
^
In considering the character of the motion denned by (11) in the immediate vicinity of the origin we see that if a < |TT, the term in r2 preponderates even when n= 1. When a— \TT exactly, the second term in (11) and the first term under £ corresponding to n = 1 become infinite, and the expression demands transformation. We find in this case
= ir2 + — (0 - ITT) cos 20 + r2 sin 2(9 (-"- log r - ^-\ +
7T                                                              \7T                  47T/
7T
......... (12)
the summation commencing at n = 3.    On the middle line Q = £TT, we have
The following are derived from (13) :
r	-^	r	-^	?•	-\tf^
o-o	•ooooo	0-4	•14112	0-8	•13030
0-1	•02267	0-5	•16507	0-9   :	•07641
0-2	•06296	0-6	•17306	i-o	•ooooo
0-3	•10521	0-7	•16210		
The maximum value occurs when r = '592.    At the point r = '592, Q = ITT, the fluid is stationary.
A similar transformation is required when a = 3?r/2.
When a = TT, the boundary becomes a semicircle, and the leading term (n = 1) is
= -!:2/---> ..................(14)
which of itself represents an irrotational motion.
K. VI.id body.    The motion expressed that which would ensue if the rotation of the vessel were suddenl; A related problem was solved a long time since by Stokes*, who c the irrotational motion of fluid in a revolving sector.    The solution problem is derivable from (10) by mere addition to the latter of i/r0 for then ^ + ^r0 satisfies V2 (^r + ^r0) = 0 ; and this is perhaps th«